college algebra

Question

Starting with the graph of , write the formula for the function that results from(a) shifting  1 units downward.    (b) shifting  1 units to the right.    (c) reflecting  about the x-axis.  

Answer 1

Greetings Kassidy,

given the function f(x)=2x I usually start by recognizing what type of function is given. In this case is a linear function (i.e. no exponents, no trig, only one term and if you were to plot points and graph it you would see that it's just a line with slope of 2).

Then, a) to shift vertically up/down you add/subtract the number of units we want to shift "outside" of the function so you would rewrite it as y=2(x) (I write parentheses around the type of function we have, in this case we established it's linear because it's just x and 2 is just the slope/factor). Then, because we want to shift it down 1, then we subtract (down) 1 "outside of the function" and the new function is y=2(x) - 1 or just y=2x -1

b) asks to shift f(x) horizontally, so in this case it's a bit tricky because if we want to shift left then we add "inside" of the function and if we want to shift it to the right we subtract "inside" of the function by the number of units we want the shift. Hence, f(x)=2(x) shifted to the right 1 unit can be written as f(x)=2(x-1) and if you were to simplify, you can distribute the 2 and get f(x)=2x -2 and both functions are equivalent, thus, the same graph.

c) for a reflection across the x-axis you would just make the function negative (or put a negative in front of f(x))

y=f(x) reflected across the x-axis -> y= - f(x). Therefore, in this case y= -2x.

I hope this works! Also, FYI these formulas work for all functions in mathematics (: